invited paper the ion trap quantum information...

m2 May 1997 MS No. APB1062

Appl. Phys. B 64, 623–642 (1997)

C© Springer-Verlag 1997

Invited paper

The ion trap quantum information processorA. Steane

Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, England(E-mail: [email protected])

Received: 16 October 1996/Revised version: 2 April 1997

Abstract. An introductory review of the linear ion trap isgiven, with particular regard to its use for quantum informationprocessing. The discussion aims to bring together ideas frominformation theory and experimental ion trapping, to providea resource to workers unfamiliar with one or the other of thesesubjects. It is shown that information theory provides valuableconcepts for the experimental use of ion traps, especially errorcorrection, and conversely the ion trap provides a valuablelink between information theory and physics, with attendantphysical insights. Example parameters are given for the caseof calcium ions. Passive stabilisation will allow about 200computing operations on 10 ions; with error correction thiscan be greatly extended.

PACS:42.50.Vk; 89.70.+c; 32.80.Pj

This paper is a contribution to the rapidly developing fieldof quantum information theory and experiment. Quantum in-formation is an interdisciplinary subject, in which computerscientists and other experts in the theory of classical informa-tion and computing are not necessarily familiar with quantummechanics, and physicists and other experts in quantum theoryare not necessarily familiar with information theory. Fur-thermore, whereas the field has enjoyed a rich theoreticaltreatment, there is a lack of an experimental basis to under-pin the ideas. This is especially significant to the issue of errorcorrection, or more generally any stabilisation of a quantumcomputer, which is among the most important unresolved is-sues in this field. The aim of this paper is to offer an aid topeople from different sides of the subject to understand issuesin the other. That is to say, the ideas of quantum informationand computing will be introduced to experimental physicists,and a particular physical system that might implement quan-tum computing will be described in detail for the benefit oftheoreticians. I hope to give sufficient information to formmore or less a ‘blueprint’ for the type of quantum informationprocessor currently achievable in the lab, highlighting the var-ious experimental problems involved. The discussion is likea review in that it brings together the work of other authorsrather than provides much original material. However, an ex-haustive review of the wide range of subjects involved is not

intended, and as a result it will not be possible to do justiceto the efforts of the many people who brought the experi-mental and theoretical programmes to their present state ofaccomplishment.

The plan of the article is as follows. In Sect. 1 the con-cepts of quantum information processing are introduced. InSect. 2 the general requirements for realising an experimen-tal processor, making use of a ‘universal’ set of quantum logicgates, are described. In Sect. 3 the linear ion trap is consideredas a system in which these ideas can be applied. A physicalprocess by which quantum logic gates may be applied in anion trap is described in detail. Limitations on the size of theprocessor (number of quantum bits) and speed of operation(‘switching rate’) are discussed. In Sects. 4 and 5 the main ex-perimental techniques required to realise the ion trap processorin the lab are discussed; these are laser cooling of the ions,and low-noise generation of the correct direct current (dc) andradio frequency (rf) voltages for the trap electrodes, as wellas a good choice of electrode design. In Sect. 6 we begin toestablish definite values for the experimental parameters, byconsidering specific candidate ions to which the methods canbe applied. Example values are given for the singly-chargedcalcium ion. In Sect. 7 experimental limitations such as un-wanted heating of the ion motion are discussed. This leads toan estimate for the maximum number of unitary operations(quantum gates) that could be carried out in the processorbefore the coherence of the system is destroyed. It is foundthat for an example case of around 10 ions, a few hundredoperations represents a severe experimental challenge. Theuse of quantum error correction to enhance the performanceis then discussed. This should allow great increases in thenumber of operations, while preserving coherent evolution.The conclusion outlines the most important avenues for futureinvestigation.

1 Quantum information and computing

Quantum information theory is concerned with understand-ing the properties of quantum mechanics from an informationtheoretic point of view. This turns out to be a very fruitfulapproach, and leads naturally to the idea of information pro-


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cessing or computing, so that one poses the question “what arethe possibilities for, and the limitations of, information pro-cessing in a physical system governed by the laws of quantummechanics?” A good deal of theoretical insight into this ques-tion has been gained; introductory discussions may be foundin [3, 4, 8, 24]. For instance, it is possible to identify a smallset of ‘building blocks’; if they could be realised and many ofthem combined, a ‘universal quantum computer’ could be con-structed. The computer is ‘universal’ in the sense that it couldsimulate, by its computations, the action of any other com-puter, and so is more or less equal to or better than any othercomputer [1]. The phrase ‘more or less equal’ has a techni-cal definition which will be elaborated in Sect. 3.3. A specificset of such building blocks is a set of two-state systems (thinkof a line of spins), and a simple unitary interaction that canbe applied at will to any chosen small set of these two-statesystems [2, 3]. In this context it is useful to describe the inter-action in terms of its propagatorU = exp(iH∆t/~) rather thanits HamiltonianH . Here∆t is some finite interval of time (one‘clock period’ in computing terminology) at the end of whichthe propagator has had just the effect desired on the computer.After this time the interactionH falls to zero (is turned off).Such a propagator is referred to as a ‘gate’, by analogy witha logic gate in a classical computer.

For quantum information processing, these requirementsmay be summed up as the need for a system (‘quantum com-puter’ or QC) with a Hilbert space of sufficient number,D,of dimensions, over which you have complete experimentalcontrol. That is, you can tell your system to go from any ofits states to any other, without uncontrollable error processessuch as relaxation and decoherence. Also, one must be able toprepare the initial state and measure the final state.

It is usual to consider a Hilbert space whose number ofdimensions is a power of 2, i.e.D = 2K , in which case wesay we have a system ofK quantum bits or ‘qubits’. Exam-ples of qubits are the spin state of an electron (2 orthogonalstates and so a single qubit) the polarization state of a photon(a single qubit), and the internal state of an atom having twoenergy levels of total spin1 and2 (8 states and so 3 qubits).Although these are all equivalent from the point of view of theproperties of Hilbert space, they are very different from thepoint of view of experimental implementation. The use of theword ‘qubit’ rather than ‘two-state system’ emphasizes thisequivalence between otherwise very different quantum sys-tems. In fact, the idea of a qubit has further significance, sinceit can be shown [5, 6] that the essential properties of any quan-tum state of any system can be transposed (by interactionsallowed by the laws of physics) into the properties of a finiteset of qubits and back again [7]. The important point is thatthe average number of qubits required to do this is equal to thevon Neumann entropy of the initial state (“quantum noiselesscoding theorem”, also referred to as “quantum data compres-sion” [8]). Therefore the qubit gives a measure ofinformationcontent in quantum systems, and is thus the correct quantumequivalent of the classical bit.

With the invention of a new word for the quantum two-statesystem accepted, there is justified resistance to the adop-tion of the terms ‘computer’ and ‘computing’ to describe thelarger quantum systems with which we are concerned. This isbecause it is an open question whether a true quantum ‘com-puter’ could ever function, since once the physical systemhas sufficient degrees of freedom to be meaningfully called

a ‘computer’, the large-scale interference necessary for paral-lel quantum computing (see below) may always be destroyedin practice, owing to the sensitivity of such interference to de-coherence. For this reason, the more modest term ‘informationprocessor’ is used here as much as possible. The ‘processing’might consist of quite simple manipulations, such as allowingone qubit to interact with another, followed by a measurementof the state of the second qubit. Even such a simple operationhas a practical use, since it can be used for error detection atthe receiving end of a quantum communication channel, lead-ing to the possibility of secure quantum key distribution forcryptography [9–11].

Decoherence and dissipation in quantum mechanics isa subject in its own right, and has been discussed since thebirth of quantum theory. Recent reviews and references maybe found in [12–14]. Its impact on quantum computers in par-ticular has been considered [15–17], and will be taken intoaccount in Sect. 7.

Let us temporarily neglect decoherence and all sources ofexperimental imprecision, and suppose that we are able to pre-pare a quantum system, of, say, two hundred qubits, and driveit through any prescribed evolution. Such an experimental ap-paratus would be able to address many interesting questionsin physics, but let us concentrate on questions of informationprocessing. Can such a system perform information process-ing in new (and hopefully powerful) ways? That is, in wayswhich are ruled out foranycomputer based on standard bina-ry logic bits 0 and 1, rather than quantum states, i.e. qubits?The answer to this question isyes. Exactly what is the essenceof quantum, as opposed to classical, information processingis still not fully clear, but it appears to depend on quantumsuperposition, entanglement, and interference. It would makethis paper too long to devote much space to this lengthy sub-ject; the reader is referred to [3, 4, 24]. However, a few wordsare in order.

A classical computer is given an input, which we can imag-ine without loss of generality to be a binary numberx.Analgorithm A prescribes various operations to be performed,resulting in an outputA(x). In the course of the computation,the computer uses an internal memory in which intermedi-ate results are stored, typically requiring more memory spacethan would be needed to store justx or A(x) alone.

Similarly, a quantum computer is prepared in an input state|x〉, and driven towards an output state|A(x)〉 (the notationadopted here is slightly over-simplified, but this will not affectthe point I wish to make). A feature of a quantum computer isthat in the course of the computation, superposition states suchas|y〉+ |z〉 are involved. It is significant that here the quantumstate ‘stores’, after a certain fashion,bothvaluesy andz, with-out using any more physical resources than those required tostore eithery or z alone. However, the same could be said forthe idea of using classical fields for information storage (forexample in a hologram): they also can support superposition.The quantum computer has a further subtlety. With 200 qubits,there are in total2200 dimensions in Hilbert space. This meansthat a 200-qubit computer can store, in a quantum superpo-sition of mutually orthogonal states,2200 different numbers.The important point is that this is ahugeeffective storagespace: it would be quite impossible to manipulate hologramsor classical fields with this number of elements, or indeed tobuild an electronic computer with this size of memory (evenwith one atom per memory bit, there are not sufficient atoms


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in the whole of the Earth!). The next point is that a singleoperatorU applied to a system in such a superposition willevolve all elements of the superpositionsimultaneously,sinceU(|y〉+ |z〉)= U|y〉+U|z〉. These properties together are re-ferred to asquantum parallelism. They hint at the possibilityof a huge speed-up in execution time for well-designed quan-tum algorithms, compared to anything possible on a classicalcomputer. It is significant also that such exponentially large su-perpositions can be produced in the quantum computer’s stateby means of a managable number (here, 200) of rotations ofthe states of individual qubits, so the large superposition is notbought at the cost of large numbers of operations to produce it.

There is a drawback, however. The final result of an al-gorithm is a single number, not2200 different numbers, sowe require our quantum algorithm to be able to bring to-gether its huge number of intermediate results. This relies onquantum interference. The difficulty of implementing quan-tum computation comes down to the fact that we rely onan interference among a huge number of different states,but such interferences are very sensitive to experimentalimprecision.

If a computational task can be framed in such a wayas to take advantage of quantum parallelism, and pro-duce an output that depends on a quantum interferencebetween the exponentially large number of intermediate re-sults, then a great speed-up in execution time is obtained,compared to any computer that cannot use such methods(i.e. any classical computer). A computational task thatis particularly amenable to this approach is that of find-ing the period of a function that is simple to evaluate,but whose period is very long, and cannot be deducedby any quicker means than evaluating the function onmany inputs. It was by reducing the task of factorisinga large integer to this form that Shor showed that an idealquantum computer can solve the important factorisationproblem [18].

Algorithms like that of Shor represent an important insightinto the nature of quantum mechanics, but it must be remarkedthat the existence of such an algorithm does not in itself implythat a quantum computer capable of running it can be built,since in the discussion above we temporarily neglected ex-perimental imprecision and decoherence. An interesting pointthus emerges: does nature actually prevent the realisation ofefficient quantum algorithms like Shor’s, not directly, but viathe ‘back door’ of decoherence? If one chooses any systemthat might support quantum computation, and makes reason-able estimates of the rate of decoherence, through thermaleffects, spontaneous emission, or experimental imprecision,a rough calculation will show that Shor’s algorithm cannotsucceed on interesting cases, i.e. factorising large numbers(> 10100). This has been emphasised by Haroche and Rai-mond [19]. However, a two-fold attack is underway on thisproblem. First, one may search for simpler quantum algo-rithms, by which even a small quantum computer, of, say, 25qubits, might run important computations. This search is sofar unfruitful, but it is interesting to note that 25 qubits iscurrently the limit on the size of a quantum system that canbe thoroughly simulated by classical computers. Second, onemay seek ways to make the quantum computer more robust.Here there has been considerable success, based on the ideaof quantum error correction[65–67, 70–72]. This is a newconcept, which will be discussed in Sect. 7.1.

2 Minimum requirements for quantum informationprocessing

It can be shown [20] that to produce arbitrary unitary trans-formations of the state of a set of qubits, which is what onewants for information processing, it is sufficient to be able toproduce arbitrary rotations in Hilbert space of any individualqubit, i.e. the propagator

exp(−iθθθθ ·σσσσ /2)=

(cos(θ/2) −e−iφ sin(θ/2)

eiφ sin(θ/2) cos(θ/2)

), (1)

and to be able to carry out the ‘controlled-rotation’ operationCROT = |00〉〈00|+|01〉〈01|+|10〉〈10|−|11〉〈11|between anypair of qubits. The notation used here is standard, the kets|0〉and |1〉 refer to two orthogonal states of a qubit. This ba-sis is referred to as the ‘computational basis’, since this aidsin designing useful algorithms for the QC. From a physicalpoint of view, it is useful to take the computational basisto be the ground and excited eigenstates of the Hamilto-nian of the relevent two-level system, though this is by nomeans required and any basis will serve. States such as|01〉are product states|0〉= |0〉⊗| 1〉 where the first ket refersto one qubit, and the second to another. For our purpos-es the qubits will always be distinguishable so we do notneed to worry about the symmetry of the states (with re-spect to exchange of particles) and any related quantumstatistics.

As mentioned previously, an operation likeCROT is a prop-agator acting on the state of a pair of qubits. In matrix form itis written

UCROT =

11

1−1

(2)

in the basis|00〉,|01〉,|10〉,|11〉, where matrix elements thatare zero have not been written. The appellation ‘controlledrotation’ comes from the fact that if the first qubit is in thestate|0〉, CROT has no effect, whereas if the first qubit is in thestate|1〉, CROT rotates the state of the second by the Pauliσ zoperator.

The two operators just described form a universal set,which means that any possible unitary transformation can becarried out on a set of qubits by repeated use of these operatorsor ‘quantum gates’, applied to different qubits [20]. Anothercommonly considered quantum gate is the ‘controlled not’ or‘exclusive or’ (XOR) gate

UXOR =

11

0110

; (3)

see also (16). This gate has no effect if the first qubit is inthe state|0〉, but applies aNOT operation (σ x Pauli spin op-erator) to the second qubit if the first is in the state|1〉.Inthe computational basis, this means that the state of the sec-ond qubit becomes theXOR of the two input qubit values. Wehave introducedCROT beforeXOR in this discussion, goingagainst standard practice, because we shall see later thatCROTis easier to implement in an ion trap.


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It should be emphasised that this model in terms of quan-tum gates operating on quantum bits is by no means the onlyway to think about quantum computation, but is the way thatis best understood at present, and is certainly very powerful.The next most studied model is currently that of cellular au-tomata [21], but the field is still young and further models willno doubt be developed in the future.

A further simplification of the physical construction ofa quantum computer is as follows. Instead of seeking a meansto carry outCROT between any pair of qubits directly, it issufficient to have one special qubit that can undergoCROTwith any of the others. This special qubit acts as a one-bit‘bus’ to carry quantum information around the computer,mak-ing repeated use of theSWAP operation|00〉〈00|+ |10〉〈01|+|01〉〈10|+ |11〉〈11|. To carry outCROT between any pair ofqubits x and y, one makes use of the bus bitB as follows:CROT(x,y)= SWAP(B,x) ·CROT(B,y) · SWAP(B,x). The oper-ation SWAP can be built out of threeXORs with the orderof the bits alternating:SWAP(B,x)= XOR(B,x) · XOR(x,B) ·XOR(B,x). However, in practice this construction is unneces-sarily complicated, sinceSWAP can be applied more or lessdirectly in most physical implementations.

The use of a bus bit makes the physical construction ofa quantum information processor much simpler, and indeedmost current proposals use this concept. However, it has themajor disadvantage that more than one gate (acting on differ-ent sets of qubits) cannot be carried out simultaneously (i.e. inparallel), except single qubit rotations. If we accept this limi-tation, the minimum requirement for our processor is arbitraryrotations of any single qubit, plusCROT and SWAP betweenthe bus qubit and any of the others. This is the minimum set of‘computing operations’, in the sense that arbitrary transforma-tions can be carried out by means of this small set. However,this establishes neither that arbitrary transformations can becarried outefficiently, nor that they can be carried outwithoutuncorrectable errors, both of which are important additionalconsiderations for a computer. We will return to these issuesin Sects. 3.3 and 7.1.

A further ingredient for quantum information processingis that the result of the process – here the final state of thequantum system – must be able to be measured without er-rors. A basis is chosen (typically the eigenbasis of the systemHamiltonian) and a measurement of all the qubits is carriedout in this basis.

To make a modest processor (a few qubits) the easiest ap-proach is probably to use single particles with several internaldegrees of freedom. Examples are a spinJ = 2K−1−1/2 ina magnetic field (sayJ = 7/2 giving 2J+1 = 8 dimensionsand thereforeK = 3 qubits); a molecule or confined particlewith 2K accessible vibrational states (‘accessible’ in this con-text means the experimenter can cause computing operationsamong the states at will). This approach will be interestingin the short term. However, it is difficult to imagine it beingextended in the longer term to enable the realisation of a re-ally interesting processor with hundreds of qubits. Also, it isnot clear how to apply arbitrary operations to a single par-ticle (spin, molecule) with an evenly spaced ladder of energylevels, owing to level degeneracies in the interaction picture.

There are now several proposed physical systems thatmight one day make a quantum computer [22–26]. We willconcentrate on the system of a line of ions in an ion trap, sinceit appears to be the most promising at present. However, de-

velopments in solid-state physics may overtake us, and oneshould bear this in mind. It is not easy to couple the quan-tum information out of an ion trap system (i.e. in the formof qubits, not classical measurements), which is important forquantum communication. In this regard the approach basedon strong coupling between an atom and a cavity mode ap-pears more useful, since there a bit of quantum informationcould in principle be transferred into the polarisation state ofa photon, which then leaves the system in a chosen direction(a ‘flying qubit’) [27]. However, such ideas could be appliedto trapped ions, making a form of hybrid processor, so the iontrap system remains an interesting candidate even for quantumcommunication purposes.

3 Ion trap method

For reviews and references on ion trapping, see for ex-ample [28–32]. The ion trap system that interests us usesa line ofN trapped ions. Each ion has two stable or metastablestates, for example two hyperfine components of the electron-ic ground state (which usually requires an odd isotope), ortwo Zeeman sublevels of the ground state, separated by ap-plying a magnetic field. The ground state and a metastableelectronic exited state (e.g. a D state for ions of alkaline earthelements) might also be used, but this is a poor choice sincethe laser linewidth and frequency, as well as most of the mir-rors etc. on the optical bench, will have to be very preciselycontrolled for such an approach to work. There have been op-timistic estimates of the computational abilities of an ion trapprocessor, based on the use of such optical transitions, but oneshould beware of the lack of realism in such estimates. Thiswill be discussed more fully once we have seen exactly howthe system is intended to operate.

There areN laser beam pairs, each interacting with one ofthe ions, (or a single beam which can be directed at will to anychosen ion), see Fig. 1. Each ion provides one qubit, the two-dimensional Hilbert space being spanned by two of the ion’sinternal energy eigenstates. A further(N+1)th qubit acts asa ‘bus’ enabling the crucialCROT operations. This qubit is thevibrational motion of the whole ion string in the trap potential.This motion must be quantised, in other words the ion cloudtemperature must be reduced well below the ‘quantum limit’defined by the axial vibrational frequency in the ion trap:

kBT ~ωz. (4)

The first major experimental challenge (after making a trap andcatching your ions) is to cool the ions down to this quantumregime. Note that the quantum regime for the trapped motionof the ion isnot related to the “Lamb–Dicke” regime, whichwill be considered below. In brief, it will be shown that onewants to operate well into the quantum regime, but on theborder of the Lamb–Dicke regime.

So far the quantum regime has been achieved only for a sin-gle ion of either mercury in two dimensions [36] or berylliumin three dimensions [37]. Both experiments used optical side-band cooling in the resolved-sideband (tight-trapping) limit.This and other possible cooling techniques will be discussed.Traps for neutral atoms have also attained the motional groundstate, most spectacularly in the case of Bose Einstein conden-sation [39], but also in optical lattices [40]. These systems


627

+

+

+

+

2.5 mm

Fig. 1. Experimental arrangement. A line of three ions sits between cylin-drical electrodes, here seen sideways on. Pairs of laser beams excite Ramantransitions, which impart momentum changes to the ions along the axial di-rection of the trap. The double-ended arrow indicates the direction of theresulting oscillations, it can be regarded as a pictorial representation of thefourth ‘qubit’ in the system (see text, Sect. 3.2). The electrodes are split inorder to allow a constant voltage to be applied between their ends, so that anaxial potential minimum occurs in the region where the long electrode seg-ments overlap. Radial confinement is provided by alternating voltages (seetext, Sect. 5)

do not (at present) provide full control of individual atomsand interactions between pairs, so we will not discuss them.However, they lend further weight to the impression that it isin atomic physics and quantum optics, rather than solid-statedevices, that quantum information processing will be mostfruitful in the immediate future.

To get to the quantum regime, it appears to be neccessary touse a Paul rather than Penning trap, since rf technology allowstighter confinement than does high-magnetic-field technology.Therefore only the Paul trap (rf trap) will be considered fromnow on, although we may permit ourselves to add a magneticfield if we wish, for some other reason such as to enhancethe stability or split the Zeeman levels. In any case, tighterconfinement enables a faster ‘switching-time’ for quantumgates such asCROT, so, as a general rule, tight traps are thebest option, though there are some qualifications to this rule,which are discussed in Sect. 3.4.

Note that if several ions are in a three-dimensional rf trapof standard geometry (with the rf voltage between end capsand a ring), then matters are complicated since no more than

one ion can be at the centre of the trap potential. Away from thecentre, ions undergo rf micromotion and this causes heatingif there is more than one ion, owing to collisions (Coulombrepulsion) which force the micromotion out of quadraturewith the rf field. To avoid this, one must use a linear or ringgeometry. The confinement along the axis is then either dueto a static field from end cap electrodes (linear case), or torepulsion between ions combined with their confinement toa ring shape. In this case, only radial micromotion is present,but this vanishes for all the ions if they lie along the axis atthe centre of the radial potential, so rf heating is avoided. Thering case must imply a small micromotion tangential to thering, since the tangential and radial confinement can not becompletely decoupled, but as far as I know this has not yetbeen found to be a problem.

3.1 Average motion

We will model a row ofN ions in a trap as a system ofN point charges in a harmonic potential well of tight radialconfinement, i.e.ωx,ωyωz; see Fig. 2. The oscillation fre-quenciesωx,ωy, andωz are parameters that will be obtainedfrom the electrode geometry and potentials in Sect. 5. Thetotal Hamiltonian is

H =N

∑i=1

12

M

(ω2

xX2i +ω2

yY2i +ω2

zZ2i +

P2i

M2

)+

N

∑i=1

∑j>i

e2

4πε0 |Ri−Rj |.

(5)

Forωx,ωyωz and at low temperatures, the ions all lie alongthe z axis, so we can take|Ri−Rj | ' |Zi−Zj |, and the radialand axial motion can be separated. The axial motion interestsus, so the problem is one-dimensional. A length scale is givenby

zs =

(e2

4πε0Mω2z

)1/3

, (6)

010

2030

40

0

10

20

30

400

0.5

1

1.5

2

Fig. 2.Schematic illustration showing an anisotropic harmonic potential withthe positions of three trapped ions indicated


628

−3 −2 −1 0 1 2 30

1

2

3

4

5

6

7

8

9

10

z

N

Fig. 3.Equilibrium positions for a line of point charges in a quadratic potential,as a function of the number of chargesN. The positions are in units ofz0defined by (6). The curve is Eq. (9)

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

10

frequency

N

Fig. 4. Normal mode frequencies for a line of point charges in a quadraticpotential, as a function of the number of chargesN. The frequencies are inunits ofωz

which is of the order of the separation between the ions(typically 10 to 100µm). Solving the classical equationsof motion (i.e. the operatorsZ, Pz become classical vari-ables z, pz), one obtains the equilibrium positions shownin Fig. 3. With more than two trapped ions, the outer ionstend to push the inner ones closer together, so the ion po-sitions depend onN [see (9)]. Remarkably, however, thefrequencies of the first two normal modes of oscillation aboutthese equilibrium positions are independent ofN (for smalloscillations) [25], and those of higher modes are nearly in-dependent ofN. The frequencies of the first two modes areωz and

√3ωz, and those of higher modes are given approx-

imately by the list1,√

3,√

29/5, 3.051, 3.671, 4.272,4.864, 5.443, 6.013,6.576, which gives the frequency ofthe highest mode, in units ofωz, for N = 1 to 10. The nearindependence ofN of the mode frequencies is illustratedby Fig. 4.

The lowest mode of oscillation corresponds to harmonicmotion of the centre of mass of the ion string. In this mode,all the ions move to and fro together. It is important that thefrequency of this mode is significantly different from that ofany other mode, since this means that experimentally one canexcite the centre-of-mass mode without exciting any of theothers.

We can now proceed directly to a quantum mechanicaltreatment, simply by treating the centre-of-mass coordinatezcm as a harmonic oscillator. The classical result that thecentre-of-mass normal mode has frequencyωz remains valideven though the ion wavefunctions may now overlap, since allthe internal interactions among the ions cancel when one cal-culates the centre-of-mass motion. Since we have an oscillatorof massNM and frequencyωz, the energy eigenfunctions are

ψ n (zcm) =

(NMωz

π~22n(n!)2

)1/4

Hn

(zcm

√NMωz/~

)×e−NMωzz2

cm/2~.

(7)

The spatial extent of the Gaussian ground state probabilitydistribution is indicated by its standard deviation

∆zcm =√~/2NMωz . (8)

Since we want a different laser beam to be able to addresseach of the ions, we require∆zcm to be small compared tothe separation between ions. The closest ions are those atthe centre of the line. A numerical solution of (5) yields thefollowing formula for the separation of the central ions:

∆zmin' 2.0zsN−0.57. (9)

This formula is plotted forN ≤ 10 in Fig. 3. An approx-imate analytical treatment forN 1 does not predicta power-law dependence of∆zmin on N, but rather∆zmin ∝ zs(log(N)/N2)1/3 [42]. However, (9) is more accu-rate for N < 10 and remains accurate for the range ofNthat interests us (up to, say,N = 1000). Setting∆zcm ∆zminyields

ωz

M

32N1.86

~3

(e2

4πε0

)2

' 2.4×1021 N1.86 Hz/u, (10)

where u is the atomic mass unit1.66057×10−27kg. Thiscondition is easily fulfilled in practice, withωz no greater thanaGHz, andM between 9 and200 u. Therefore it is legitimateto picture the ions as strung out in a line, each sitting in a smallwavepacket centred at its classical equilibrium position, notoverlapping the others. Note that (10) does not guarantee thatthe ions are sufficiently separated to be addressed by differentlaser beams, only that their wavefunctions do not overlap.

In the above, it was assumed that the radial confinementwas sufficient to cause the ions to lie along thez axis, ratherthan form a zigzag or helix about it. The onset of suchzigzag modes has been studied numerically [41] and ana-lytically [42]. They occur when the ions approach sufficientlyclosely that the local potential minimum at the position of anion on thez axis becomes a saddle point. For a string of ionsuniformly spaced by∆z (which is not the case in our harmon-ic trap), the transition from a line to a zigzag occurs when


629

ω2r ' 4.2072(zs/∆z)3ω2

z [43], where we have taken the caseωx = ωy≡ω r . Setting∆z= ∆zmin leads to the condition

ω r

ωz> 0.73N0.86 (11)

for the prevention of zigzig modes. ForN 1, an approximateanalytic treatment yields the condition [42]

ω r

ωz> 0.77

N√

logN. (12)

These numerical and approximate analytic formulae are within10% agreement for3< N < 2000.

3.2 Principle of operation

The principle of operation of an ion trap ‘information pro-cessor’ was described by Cirac and Zoller [25], and the mostimportant elements of such a system were first realised inthe laboratory by Monroe et al. [44]. Whereas the transi-tion operators given by Cirac and Zoller were calculated forstanding-wave excitation of allowed single-photon transitions,experimentally Monroe et al. employed travelling-wave exci-tation of two-photon Raman transitions (cf Figs. 1 and 8). Thebasic form of the operators is independent of the type of ex-citation used, however. The method may be understood byreference to Fig. 5, which shows the relevant energy levels forone of the ions in the trap. We consider three of the ion’s inter-nal energy eigenstates|F1,M1〉, |F2,M2〉 and|Faux,Maux〉, andvarious excitations of the centre-of-mass motion. The ion’s in-ternal energy levels are separated in frequency byω0 andωaux

1,0

1,1

0,0

0,1

aux,0

aux,1

F1,M1

F2,M2

Faux,Maux

wz

w0

waux

XOR

SWAP

Fig. 5.Energy levels and transitions in a single ion significant for informationprocessing with a line of trapped ions. The labelsF , M indicate differentinternal states of the ion. Each internal state has an associated set of vibrationallevels for each of the vibrational modes. Here, just the ground and first levelsof the lowest mode (spacingωz), are shown in the main diagram, and theinsert shows the further low-lying vibrational levels whose excitation wewish to avoid. The full arrows indicate transitions at frequenciesω0−ωz andωaux+ωz, which are used in theSWAP(−i) andCROT operations described inthe text. Note that radiation at a given frequency couples not only the levels atthe two ends of the relevant arrow on the diagram, but also other pairs of levelswith the same difference of vibrational quantum number. The figure showsωaux to be of the same order asω0, because this is what typically occurs whenalkali-like ions are used. The vibrational frequency is smaller,ω0 ' 1000ωz

as indicated on Fig. 5. Note that all these levels are low-lying,separated from the ground state only by hyperfine and Zee-man interactions (see Fig. 8), so their natural lifetime againstspontaneous emission of rf photons is essentially infinite. Fig-ure 5 shows the lowest-lying excitations of the second, third,and fourth normal modes as well as the first, to act as a re-minder of the location of the closest extraneous levels whoseexcitation we wish to avoid. The energy eigenstates of thevibrational motion may be written as|n1,n2,n3, . . .〉, wheretheni are the excitations of the various normal modes. Onlythe ground state|0,0,0, . . .〉 and first excited state of the cen-tre of mass|1,0,0, . . .〉 will be involved in the operations wewish to invoke. This centre-of-mass vibrational degree of free-dom is often referred to somewhat loosely as a ‘phonon’. The‘computational basis’ consists of the states

|0,0〉 ≡ |F1,M1〉⊗ |0,0,0, . . .〉 ,

|0,1〉 ≡ |F1,M1〉⊗ |1,0,0, . . .〉 ,

|1,0〉 ≡ |F2,M2〉⊗ |0,0,0, . . .〉 ,

|1,1〉 ≡ |F2,M2〉⊗ |1,0,0, . . .〉 .

(13)

It will now be shown how to carry outCROT between anysingle ion’s internal state and the bus (phonon) bit, then howto carry out arbitrary rotations of the internal state of an ion,then how to carry outSWAP between any ion and the bus bit.From the discussion in Sect. 2, these three operations forma universal set and so allow arbitrary transformations of thestored qubits in the processor.

The auxillary states|aux,i〉 ≡ |Faux,Maux〉⊗ |i,0,0, . . .〉(i = 0,1) are available as a kind of ‘shelf’ by means ofwhich useful state-selective transformations can be carriedout among the computational basis states. If one applies ra-diation at the frequencyωaux+ωz, then inspection of Fig. 5will reveal that only transitions between|1,1〉 and|aux,0〉willtake place (if we assume that unwanted levels such as|1,2〉are unoccupied).1 If one applies a2π pulse at this frequency,then the state|1,1〉 is rotated through2π radians, and thereforesimply changes sign. In the computational basis, the effect isequal to that of theCROToperator described in Sect. 2, see (2).

A 2π pulse at frequencyω0−ωaux−ωz also produces a con-trolled rotation, only now the minus sign appears on the secondelement down the diagonal of the unitary matrix, rather thanon the fourth, causing a sign change of the component|0,1〉rather than of|1,1〉. This case will be calledC¬ROT, the nega-tion symbol¬ referring to the fact that here the second qubitis rotated if the first is in the state|0〉 rather than|1〉.

To rotate an ion’s internal state without affecting the centre-of-mass motion, one applies radiation of frequencyω0. If suchradiation has phaseφ with respect to some defined origin ofphase, and duration sufficient to make apπ pulse, then theeffect in the computational basis is

Vp(φ)≡

(cos(pπ/2) −ie−iφ sin(pπ/2)

−ieiφ sin(pπ/2) cos(pπ/2)

)ion⊗

(1001

)cm

,

(14)

1Cirac and Zoller originally proposed to produce the selective effect of thisCROToperation by means of a chosen laser polarisation rather than frequency.However, frequencies can be experimentally discriminated more preciselythan polarisations, which explains why Monroe et al. chose to use a frequency-selective rather than polarisation-selective method.


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where we have followed the notation of [25], but usedp in-stead ofk to avoid confusion with the wave vector. Note thatto apply such rotations succesfully, it is necessary to have thephase of the radiation under experimental control. That meansunder control at the position of the ion, not just in some sta-ble reference cavity. This constitutes a severe experimentalconstraint, which makes computational basis states separatedby radio frequencies highly advantageous compared to statesseparated by optical frequencies. On the other hand, in orderto have the right phase experimentally, note that one need notworry about the continuous precession at frequencyω0 causedby the internal Hamiltanian of each ion. The laser field keepsstep with this precession, as becomes obvious when one usesthe interaction picture, which we have done implicitly in writ-ing (14). A possible problem arises when different ions havedifferent internal energies, owing to residual electric and mag-netic fields in the apparatus. However, this particular problemhas a fairly simple experimental solution: since each ion is ad-dressed by a different laser beam pair, one can independentlytune the laser beams driving each ion, by means of acousto-optic modulators in the beam paths. Such modulators will berequired in any case to allow the laser beam intensities to beswitched.

The centre-of-mass motion acts as the ‘bus’ qubit de-scribed in Sect. 2. To carry outXOR(B,x), between the ‘bus’and the internal state of a single trapped ion, Monroe et al.applied first aπ/2 pulse at frequencyω0,

V1/2(−π/2) =1√

2

(11−11

)ion⊗

(1001

)cm

, (15)

followed by CROT as described in the paragraph after (13),followed by a secondπ/2 pulse atω0 with phase displaced byπ with respect to the first,2 i.e.V1/2(π/2). A straightforwardcalculation shows that this sequence produces exactly

XOR(cm, ion) =

1000000100100100

, (16)

By symmetry, to obtainXOR(ion, cm), one might imagineusing a similar sequence, but with theπ/2 pulses applied atfrequencyωz so as to affect the vibrational state without af-fecting the internal state. However, the vibrational degree offreedom is not really a two-level system, so this will not work(indeed, it will cause unwanted multiple excitations of thevibrational motion). To performCROT, we made use of a tran-sition at frequencyωaux+ωz. Note that this relied on the factthat there was no population in the state|aux,1〉 (which wouldhave become coupled to|1,2〉, which is outside the computa-tional Hilbert space). This illustrates the general method bywhich the vibrational state is influenced: one uses radiation ata frequency offset from an internal resonance of the ion byωz,thus coupling levels of vibrational quantum numbers differingby 1. To avoid coupling higher-lying vibrational states, oneof the possible initial states must be unoccupied when sucha transition is invoked.

2In fact Monroe et al. state that they usedV1/2(π /2) for the first pulse, andV1/2(−π /2) for the third, producingXOR with an additional rotation of thecentre-of-mass state.

The transition at frequencyω0−ωz is indicated on Fig. 5.A moment’s reflection allows one to convince oneself thatas long as there is no population in the|1,1〉 state (nor inextraneous states such as|0,2〉), application of this radiationwill cause only transitions between|1,0〉 and|0,1〉, and hencea SWAP operation is available between the bus qubit and anyother. Applying apπ pulse at phaseφ and frequencyω0−ωz,we obtain the operation

Up(φ)≡

1 0 0 ×0 cos(pπ/2) −ie−iφ sin(pπ/2) ×0 −ieiφ sin(pπ/2) cos(pπ/2) ×0 0 0 ×

, (17)

where the final column of crosses indicates that an initial state|1,1〉 is carried out of the computational basis byUp(φ). Thecasep = 1, that is aπ pulse, produces aSWAP operation withan additional−i phase factor, which we will writeU1(0) =SWAP(i). Applying U1(0) to ion x, followed by C¬ROT toion y (i.e. using the frequencyω0−ωaux−ωz), followed byU1(0) once again to ionx, has the effect of aCROT oper-ation betweenx and y. That is,CROT(x,y) = SWAP(i)(B,x) ·C¬ROT(B,y) ·SWAP(i)(B,x), as long as the initial state ofx andthe bus is not|1,1〉. To apply the method, one uses the busas a ‘work bit’ that is arranged always to return to state|0〉before operations such asUp(φ) are applied, so the quantuminformation processing can go forward without problem.3

So far we have described operations on the ion trap bymeans ofpπ pulses. A complimentary technique is that ofadiabatic passage, in which a quantum system is guided fromone state to another by a strongly perturbing Hamiltonianapplied slowly. For example, instead of swapping one ion’sinternal state with the bus qubit, and then swapping the buswith another ion, one could swap the internal state of two ions‘via’ the bus but without ever exciting the first vibrational level.The details are described for a related system in [26]. Thismethod has experimental advantages in being insensitive tofeatures such as the timing and interaction strength. However,it can be more sensitive to off-resonant coupling to other levels,compared with the pulse technique, and this considerationled Monroe et al. to favour rf pulses. Bothpπ pulses andadiabatic passage will probably have their uses in a practicalQC (whether based on an ion trap or some other system).

The laser pulses described provide the universal set of‘quantum logic gates’ for the linear ion trap. To completethe operation of our processor or QC, we require that thefinal state of the quantum information processor can be meas-ured with high accuracy. This is possible for trapped ions bymeans of the ‘electron-shelving’ or ‘quantum-jumps’ tech-nique [28, 30, 32–34]. That is, one may measure whethera given ion is in state|F1,M1〉 or |F2,M2〉 by illuminatingit with radiation resonant with a transition from|F1,M1〉 tosome high-lying level, whose linewidth is small enough sothat transitions from|F2,M2〉 are not excited. If fluorescenceis produced (which may be detected with high efficiency), theion state has collapsed to|F1,M1〉, if none is produced, theion state has collapsed to|F2,M2〉. This method requires thatspontaneous decay from the high-lying level to|F2,M2〉 is

3Indeed, the bus may even be measured at those times when it should be inthe ground state|0〉, producing a slight stabilisation or error detection; seeSect. 7.1. In the ion trap, however, one can only thus measure the vibrationalstate by first swapping it with the internal state of a prepared ion.


631

forbidden, which is possible in this case through the angularmomentum selection rules for electric dipole radiation. Notethat typically a quantum process may need to be prepared,run, and measured several times in order to gather more in-formation about the processor’s final state than is availablefrom a single measurement of all the qubits. This is relatedto the ideas of quantum state tomography, which have recent-ly been demonstrated in an ion trap experiment (see [35] andreferences therein).

3.3 Efficient gate sequences

It was shown in the previous section thatCROT can be appliedto any pair of qubits, and arbitrary rotations of single qubitscan be carried out. Hence, as explained in Sect. 2, any arbitrarysequence of unitary transformations of the quantum processorcan be brought about. However, the most efficient methods willnot blindly adopt a simple repetition ofCROTs and rotationsto solve any problem. There may be much more efficientmethods, by using other possible pulse sequences. Cirac andZoller emphasize this by demonstrating how to apply aCnROToperation, in which theσ z operator is applied to one ion’sinternal state only ifn other ions are in the state|1〉, usinga number of pulses equal to2(n−2)+3. This is efficient inthat the number rises only linearly withn, and the multiplyingfactor is small (i.e. 2 rather than 48 as in [45]).

Efficiency in computer science has a rigorous definition.Without the details, the essential point is that if the numberof elementary computational steps (here, quantum gates) re-quired to complete an algorithm rises exponentially with thesize of the input to the algorithm, then the algorithm is ineffi-cient. The definitions can be made rigorous, which we will notattempt to do, but essentially each algorithm addresses not oneinstance of a problem, such as to “find the square of 2357”, buta whole class of problems, such as, “given an integerx, find itssquare”. The ‘size of the input’ to the algorithm is measuredby the amount of information required to specifyx, which isthe number of digits in the binary expression ofx, i.e.log2(x).A computation is inefficient if the number of steps is expo-nential inlog(x), i.e. is proportional tox. Similarly, a quantumgate involvingn qubits is inefficient if the number of physi-cal operations, such as laser pulses, required to implement itis exponential inn (e.g. increases as2n). The strict definitionof the universal computer mentioned in Sect. 1 also involvesthis efficiency aspect: when a universal computer simulatesthe action of another, the number of operations in the simula-tion algorithm must not rise exponentially with the amount ofinformation required to define the simulated computer.

Although we emphasised in Sect. 2 that a small set ofgates is ‘universal’ in that all unitary transformations can becomposed by them, this doesnot necessarily imply that theycan be used to build the particular transformations we maywant in an efficient way. In this sense, the word ‘universal’ ismisleading.

So far, networks of quantum gates have been designed forthe most part without regard to the exact physical processthat might underlie them. However, in such an approach it isnot obvious which gates to call ‘elementary’, since a physicalsystem like the ion trap may be particularly amenable to sometransformations. We have already seen an example in theSWAPgate in the ion trap, which can be carried out without recourse

to a sequence ofXOR gates. This implies that a thoroughunderstanding of a particular system like the ion trap maylead to progress in finding efficient networks. The importantinsight in Cirac and Zoller’s construction of theCnROT gate isthat the method makes use ofπ pulses at frequencyωaux+ωz.In other words, during the implementation of this gate the ionsare deliberately carried out of the computational Hilbert space.Alternatively, one could regard the ‘shelf’ level|Faux,Maux〉⊗|i,0,0, . . .〉 as within the computational Hilbert space, in whichcase we have more than one qubit available per ion. Later, inSect. 7.1, we will consider using vibrational modes in additionto the lowest one in order to have more than one ‘bus’ qubit.

3.4 Switching rate

The previous section showed how the ion trap information pro-cessor worked, by invoking radiation of prescribed frequencyand duration in the form ofpπ pulses. The ‘switching rate’ ofthe processor is limited by the duration of these pulses.

Let Ω be the Rabi frequency for resonant excitation ofthe internal transition at frequencyω0 for a free ion. Thiswill be determined by the linestrength of the transition andthe laser power available. For a two-level atom one hasΩ2 =6π Γ I/~ck3 whereΓ is the linewidth of the transition,I is theintensity of the travelling wave exciting the transition, andk isthe wavevector. When we consider excitations of the internalstate alone of an ion in a trap, i.e.∆n = 0 wheren is thevibrational quantum number, this ‘free ion’ Rabi frequencystill applies. However, when changes in the vibrational stateare involved, i.e. transitions at frequencyω0±ωz producing∆n =±1, an additional scaling factor∆zcmkz appears, where∆zcm is the extent of the ground-state vibrational wave functiongiven in (8), andkz = kcos(θ) is the wavevector componentalong thez direction. Using (8), we have

∆zcmkz =

(~k2 cos2(θ)

2NMωz

)1/2

≡η√

N, (18)

whereη is the Lamb–Dicke parameter for a single trappedion. In the case of weak excitation, the effective Rabi frequen-cy for the vibrational-state-changing transitions isηΩ/

√N,

a result that can be interpreted as arising from conservationof momentum. The factor

√N appears because the whole ion

string movesen masseand therefore has an effective massNM (Mössbauer effect). The Lamb–Dicke parameter can alsobe written in terms of the recoil energy (energy of recoil of anion after emission of a single photon)

ER≡(~k)2

2M, (19)

givingη ≡ cos(θ)(ER/~ωz)1/2.

We can now obtain a measure of the switching rateR bytaking it as the inverse of the time to bring about a2π pulseon a vibrational-state-changing transition, i.e.

R'ηΩ

2π√

N(η ≤

√N) . (20)


632

Outside the Lamb–Dicke limit4 (i.e. for η >√

N) the ion–radiation interaction is more or less equivalent to that of a freeion, so the factorη/

√N no longer applies; it is replaced by

a factor less than or equal to 1.It was remarked in the previous section that to maintain

phase control between (and during) computing operations,there is a strong advantage in having the transition frequenciesω0,ωaux in the rf to microwave rather than optical region of theelectromagnetic spectrum. However, if the relevant transitionsare driven directly by microwave radiation, with a frequencyof the order of the vibrational frequencyωz, then the Lamb–Dicke parameter is extremely small (of order(~ωz/2Mc2)1/2),so vibrational-state-changing transitions are almost impos-sible to drive. One way to avoid this would be to make the trapextremely weak, but this has the disadvantage of making thesystem sensitive to perturbations and lowering the switchingrate. Instead, it is better to drive the microwave transitions byRaman scattering at optical frequencies. This combines the ad-vantage of a large photon momentum and hence strong drivingof vibrational-state-changing transitions, with the possibilityof accurate phase control since only the phasedifferencebe-tween the pair of laser beams driving a Raman transition needbe accurately controlled. The Raman technique was adoptedfor these reasons by Monroe et al. [44]. The same reason-ing leads to the advantage of Raman scattering for preciselaser manipulation of free atoms [47, 48]. A clear theoreticalanalysis is provided by [49].

The maximum switching rate is dictated by the three fre-quenciesΩ, ωz, andER/~ in a subtle way. If only low laserpower is available,Ωωz, then the Rabi frequency limits theswitching rate and the best choice forωz is that which makesη ∼√

N, i.e.

η2 ≡cos2(θ)ER

~ωz' N . (21)

Therefore the recoil energy, given by the choice of ion andtransition, dictates the choice of trap strength, for a givennumber of ions. Typical recoil energies for an ion are in theregionER∼ 2π~× (10–200) kHz, and traps with this degreeof confinement are now standard. In this situation, increasingthe number of ions does not affect the switching rate, butreduces the required trap confinement, making the systemmore sensitive to perturbations.

If higher Rabi frequencies are available, one’s intuitionsuggests thatωz becomes the limit on the switching rate,sinceΩ must be less thanωz or the power broadening will nolonger allow the different vibrational levels to be discriminat-ed. However, at highωz one hasη 1 (Lamb–Dicke regime)so the switching rate on∆n =±1 transitions cannot reachωzif Ω < ωz. Placing thead hoclimit Ω < ωz/10 in (20), oneobtains

R<1

20π

(ERωz

~N

)1/2

. (22)

The switching rate is thus limited by the geometric averageof ωz andER/~, and the processor slows down when more

4The Lamb–Dicke limit is the condition that the ions’ motion is confined toa region small compared to a wavelength of the radiation under consideration.Since in the present case the ion string is in the ground state of the trap(quantum regime), the Lamb–Dicke limit corresponds to smallη.

ions are involved. For example, to achieve a switching at therecoil frequency, i.e.R= ER/2π~, with N = 10 ions, (22)impliesωz = 1000ER/~ andΩ = 100ER/~. To keep the ionsin a straight line, (11) requiresω r > 5300ER/~ which is veryhard to achieve experimentally.

There is another problem with increasingωz in order to in-creaseR. Whenωz is large,η

√N, so the transitions that

do not change the vibrational state (∆n = 0) are much morestrongly driven by the laser than those that do [∆n =±1, equa-tion (20)]. This increases the unwanted off-resonant drivingof ∆n = 0 transitions when∆n =±1 transitions are invoked toperform quantum gates between an ion and the phonon ‘bus’.Cirac et al. [25, 57] have emphasized the possibility of usingstanding-wave rather than travelling-wave excitation to avoidthis problem, since∆n = 0 transitions are suppressed if theion is positioned in the node of a standing wave. However, itmay not be technically feasible to achieve this for more thana few of the ions.

In principle it should be possible to run an ion trap pro-cessor at rates of orderωz by relaxing the conditionΩ < ωzand allowing off-resonant transitions, but the simple analysisgiven in Sect. 3.2 is then no longer valid. One can no longeruse a two-level model for each transition of the ion/centre-of-mass system. The ac (alternative current) Stark effect (lightshift) will be all-important, and different computational ba-sis states will become mixed by the ion–light interaction. Theoptical Bloch equations remain solvable (numerically if notanalytically), and a detailed analysis should still enable use-ful elementary computing operations to be identified. Such ananalysis is a possible avenue for future work [46].

4 Cooling

To make the quantum information processor described in theprevious sections, the main initial requirements of an experi-mental system are cooling to the quantum regime (4), andconfinement to the border of the Lamb–Dicke regime (21).The ions must be separated by at least several times the laserwavelength [see (9)], but this is automatically the case, forsmall numbers of trapped ions, since with current technolo-gy the ions are always separated by many times the width oftheir vibrational ground state wavefunction [inequality (10)],which is itself approximately equal to the laser wavelength,given that the Lamb–Dicke parameter is of order 1.

Surveys of cooling methods in ion traps are given in [31,32]. For cooling to the quantum regime, there are two possibleapproaches. Either one may cool to the ground state of a tighttrap havingη 1, then adiabatically open the trap toη ∼ 1,or one may apply cooling to a trap already atη ∼ 1. Theadvantage of the former approach is that one does not requirecooling below the recoil limitkBTR = ER. The advantage of thelatter is that strong confinement is not necessary, but attainingthe quantum regime withη ∼ 1 requires sub-recoil cooling.

Cooling to the quantum regime has so far been demon-strated for trapped ions by means of sideband cooling inthe resolved-sideband limit [36, 37]. This is described inSect. 4.1 below. However, it may be interesting to pursueother approaches, as discussed in Sect. 4.2 and Sect. 4.3.

The physics of sideband cooling is very closely related tothat involved in information processing in the ion trap. This isno coincidence, and a similar link will probably be found in all


633

physical implementations of quantum information processing.The relationship is sufficiently close that one may say thatonce the goal of laser cooling to the motional ground state isachieved in any given experimental ion trap, a primitive formof quantum information processing can proceed immediately,since all the required experimental components will be inplace. Conversely, quantum error correction (see Sect. 7.1) isa special type of ‘cooling’.

4.1 Sideband cooling

Sideband cooling is the name for the simplest type of lasercooling of a confined ion. The name comes from how thephoton-scattering process looks in the resolved-sideband limit(see below). In the case of free ions, the corresponding processis radiation pressure or Doppler cooling.

There are several significant frequencies or energies. First,we have the vibrational frequency in the ion trap potential,ωz. Next, we have the radiative width of the transition used todo the cooling,Γ . Either a single photon transition is used, inwhich caseΓ is its natural width (or possibly its broadenedwidth if another laser is used to broaden a very narrow levelas in [36]), or a stimulated Raman transition is used, in whichcaseΓ is some combination of the inverse of the duration ofthe Raman pulses, and the time for optical pumping out of oneof the states linked by the Raman transition. The physics in theRaman case and single-photon case is very similar. The Ramanmethod is a way of providing a very narrow transition whenone is not already available. It also combines the advantages ofprecise frequency control (in the rf regime) with large photonrecoil (optical regime), which permits fast cooling, for thesame reason that the switching rate for information processingis faster (Sect. 3.4). One could instead use an rf or microwavetransition, but then the cooling would be a lot slower and maynot compete well enough with heating processes.

Laser cooling of atoms is often done quite happily byusing strong, resonant transitions. Indeed, such transitionsare eagerly sought out. Why the talk of narrow transitions inthe previous paragraph? It is because simple Doppler coolingleads to the well-known Doppler cooling limitkBTD ' ~Γ /2,when the recoil energy is small compared to~Γ (this appliesin a trap as well as to free atoms). However, we want to get tothe quantum limit (4), so we require

ωz Γ . (23)

This equation is a further constraint on the performance ofthe trap. It says the cooling transition must be narrow enough,or the trap confinement tight enough, to resolve the motionalsidebands in the Lamb–Dicke spectrum.

In the resolved sideband limit, radiation pressure coolingis called sideband cooling. A nice way of understanding it isto consider it as a form of optical pumping towards the stateof lowest vibrational quantum number [31]; see Fig. 6. Notethat the recoil after spontaneous emission produces heating.The average change in the vibrational energy per spontaneousemission is equal to the recoil energy~ωz〈∆n〉= ER (a par-ticularly clear derivation of this fact may be found in [51]).For a single trapped ion illuminated by low-intensity light, the

cooling is governed by the following equation [51, 52]:

ddt〈H〉=

Iσ 0

~ωL∑n

Pn∑f

(Ef−En+ER)∣∣∣〈ψ n|eik·R|ψ f 〉

∣∣∣2×g(ωL−(Ef−En)/~) ,

(24)

where I is the intensity of the incident radiation (a singletravelling wave),σ 0 is the resonant photon scattering crosssection (σ 0 = 2π λ2 = (2π )3/k2 for a two-level atom),~ωL =~ck is the laser photon energy,Pn is the occupation probabilityof the nth energy level of the vibrational motion, of energyEn = ~ωz(n+1/2) and wavefunctionψ n [see (7)], andg(ω)is the lineshape function. For a two-level atom,

g(ω) =Γ 2/4

(ω−ω0)2 +Γ 2/4. (25)

The quantityd〈H〉/dt is the rate of change of the mean to-tal energy of the ion, averaged over an absorption/spontaneousemission cycle. Since the ion’s internal energy is left un-changed, this is the rate of change of the mean kinetic energy.Equation (24) has a simple physical interpretation as a sumof energy changes associated with radiative transitions up anddown the ladder of vibrational energy levels. At the lowest at-tainable temperature,d〈H〉/dt = 0 and one possible solution

g, 0 g, 5g, 4g, 3g, 2g, 1

e, 0 e, 5e, 4e, 3e, 2e, 1 . . .

. . .

Fig. 6. Sideband cooling. A laser excites transitions in which the vibrationalquantum number of a confined ion falls by 1 (or a higher integer). Spontaneoustransitions bring the ion’s internal state back to the ground state, with thevibrational quantum number changing by±1 or 0. On average the vibrationalquantum number is reduced, until the vibrational ground state is reached.The internal ground and excited states are|g〉 and|e〉, and the figure showsthe different vibrational levels spread out horizontally for clarity. When both|g〉 and|e〉 are long lived (for example they may be the computational basisstates: cf Figures 5 and 8), the|g,n〉 → |e,n−1〉 transition is driven by aπpulse, (U1(0) operator), and the spontaneous transition is a Raman transitionvia an unstable excited state (optical pumping). Note that such experimentaltechniques are identical to those required for information processing and errorcorrection. To cool a crystal of several ions, it is sufficient for the laser tointeract with only one ion since the Coulomb coupling between ions causesrapid thermalisation of their motional state. However, the coupling betweendifferent normal modes is weaker, so these may need to be cooled separatelyby tuning the laser to the various normal mode sideband frequencies


634

of (24) is the thermal distribution

Pn = (1−s)sn , (26)

wheres is the Boltzmann factors= exp(−~ωz/kBT), and theprobability distribution has been normalised.

At sufficiently low temperatures, all but the lowest energylevels can be ignored in (24). By usingk ·R=η(a†+a), wherea|ψ n〉 =

√n|ψ n−1〉, and expanding in powers of the Lamb–

Dicke parameter, it is a simple matter to obtain

ddt〈H〉 ' Iσ 0

ER

~ωL〈n〉 [g(ωL−ωz)−g(ωL +ωz)]

+[g(ωL)+g(ωL−ωz)] ,(27)

where 〈n〉= ∑nPn is the ion’s mean vibrational quantumnumber. Now assumeΓ ωz [inequality (23)] and let theincident radiation be tuned to the first sideband below reson-ance,ωL = ω0−ωz, then the cooling limitd〈H〉/dt = 0 leadsto a mean vibrational quantum number [52, 53]:

〈n〉 '5Γ 2

16ω2z

. (28)

Note that since〈n〉 is proportional to(Γ/ωz)2, the experimen-

tal constraint (23) will ensure achievement of the quantumlimit 〈n〉 1. This also justifies our ignoring higher energylevels in deducing (28).

The above derivation assumed a single direction of propa-gation for the cooling laser, which will result only in coolingalong one direction, so our calculation has been one dimen-sional. Taking into account the fact that spontaneous photonsare emitted into all directions, we find they do not heat anygiven dimension quite as much as we assumed, and the factor5 in (28) is replaced by(1+4α) whereα ' 2/5 depends onthe dipole radiation emission pattern [54]. However, this cor-responds to an experiment in which the motion in the otherdimensions is heated, which we wish to avoid. To cool all threedimensions, one can either introduce three laser beams, or usea single beam propagating at an oblique angle to all the prin-ciple axes of the trapping potential, and tune it separately toresonance with the three sideband frequenciesωL−ωx,y,z. Forthis one must have all three frequencies distinct, i.e.ωx 6= ωy.

It is commonly imagined that sideband cooling is not pos-sible if the recoil energy is greater than the phonon energy~ωz,since then the cooling that results from photon absorption isundone by the recoil from photon emission, andd〈H〉/dt > 0.However, one can always tune to the next lower sideband,ωL = ω0−2ωz, and good cooling is regained, as a thoroughanalysis of (24) will show. Therefore it is not necessary tobe well into the Lamb–Dicke regime in order to attain thequantum limit by sideband cooling.

Note also that both (27) and (28) are significant in order tofind the minimum temperature one will obtain in the lab. Thisis because there will always be heating mechanisms present,such as a coupling between the stored ions and thermal volt-ages in the electrodes (see Sect. 7), so it is the coolingrate,(27), not just the minimum possible temperature, which is im-portant. This has been emphasised by Eschner et al. [38] whopropose a subtle variation on sideband cooling. Their methoduses repeated measurements of the motional state by lookingfor resonance fluorescence on a sideband. The measurement

probes the high-energy part of the distribution of the ion’spopulation amongst the quantum states of the trap. The ob-servation of no fluorescence, i.e. a null detection, implies thatthe ion’s state has collapsed onto the relatively lower energypart of its initial vibrational distribution. Further such meas-urements provide opportunities for further cooling. On anygiven application of this method, the measurement may heator cool the ion, but cooling is more likely and one knows whenit has occurred. This idea is reminiscent of forced evaporativecooling (see Sect. 4.3), only here it is applied to the proba-bility distribution of a single confined particle rather than anensemble.

4.2 Sisyphus cooling

The constraint (23) means that sideband cooling will either beslow and therefore not compete well with heating processes,or will require the use of Raman transitions. We can avoidΓ ωz and nevertheless use laser cooling to get close to thequantum regime, by the use of ‘Sisyphus’ cooling [55, 56].This makes use of optical pumping and optical dipole forces(forces associated with a position-dependent ac Stark shift ofthe atomic energy levels) in a laser standing wave, on an atomwith at least three internal states. When the dipole force iscaused by a position-dependent polarisation of the standingwave, the cooling is referred to as ‘polarisation gradient cool-ing’. Theoretical analyses [55–57] have so far concluded thatthe lowest temperatures attainable by this method correspondto a mean vibrational quantum number〈n〉 ' 1, i.e. just onthe border of the regime we require. However, the coolingrate is important as well as the theoretical minimum tempera-ture, and for this reason Sisyphus cooling may be attractive forcooling a whole string of ions [58], as required for the infor-mation processor, since it is relatively fast. This would formthe precooling, which usually is necessary in any case beforesideband cooling, or something similar, can be applied to getwell into the quantum regime.

4.3 Statistical mechanical cooling methods

So far, all the cooling techniques described have been basedon laser cooling. However, for trapped neutral atoms the tech-nique of forced evaporative cooling has been shown to beextremely powerful, enabling the temperature in a weakly in-teracting atomic vapour to be brought well into the quantumregime of a trap, which for a cloud of bosons leads to BoseEinstein condensation [39].

In forced evaporative cooling, one starts with a large num-ber of trapped particles in thermal equilibrium. Those ofhigher energy are forced to leave the trap, and those remain-ing rethermalise towards a lower equilibrium temperature. Thetechnique relies on an ability to remove selectively particlesof higher than average energy. One way to do this is to reducethe depth of the trap, allowing the faster particles to fly out.Clearly this approach will work only if the thermal energy islocated more in some particles than in others, which is true fora gas of weakly interacting particles, but not for a crystalisedsystem such as a cold string of trapped ions. However, evapo-ration may be useful in an ion trap as a first stage of cooling, tobring about crystalisation. Also, it is conceivable that a Bosecondensate of neutral atoms may one day be sufficiently easy


635

to produce in the vicinity of an ion trap that it may be used asa cold reservoir to cool the ions through collisions. The useof one species to cool another is referred to as ‘sympatheticcooling’.

5 rf requirements

We now turn to the design of the ion trap itself. The elec-trode structure of the trap consists of a two-dimensionalrf quadrupole plus an axial static potential. Concentratingon the two-dimensional quadrupole, consider first the mostsimple case, in which the point in the centre of the elec-trode structure remains at zero potential, and we omit anyaxial confinement. The potential on one pair of diagonal-ly opposed electrodes is(U−V cosΩVt)/2, and that on theother pair has equal magnitude and opposite sign to this.Here ΩV is the frequency of the applied voltage, the sub-script is necessary to distinguish it from the Rabi frequencyof a driven atomic transition introduced in previous sections.The potential as a function of position in thex-y plane isφ(x,y,t) = (U−V cosΩVt)(x2−y2)/2r2

0 wherer0 is a measureof the electrode separation.5 For the case of cylindrical elec-trodes,r0 is the distance from the axis to the surface of theelectrodes [59]. The trapping effect in the radial direction isstable as long asΩV is not too small, and is strong as long asΩV is not too large. This may be parametrised in terms of thestandard parameters

a =4eU

Mr20Ω2

V

, (29)

q =2eV

Mr20Ω2

V

, (30)

wheree is the charge on a trapped ion. For present purposes,a zero dc potential differenceU = 0 may be used, soa = 0. Theradial confinement is then stable as long asq is less than about0.9 [28, 32]. The radial micromotion has a velocity amplitudeof qΩV%/2 for an ion at average distance% from thez axis.The average motion on a time scale slow compared to1/ΩV,the so-called secular motion, can be modelled in terms ofthe pseudopotential12 Mω2

r (x2 +y2)/e with radial vibrationalfrequency

ω r =√

a2 +q2/2ΩV

2=

qΩV

2√

2(a = 0). (31)

Choosingq = 1/√

2 so as to be comfortably in the zone ofstability of the trap, we obtainω r = ΩV/4. From this theLamb–Dicke parameter for the radial confinement is obtainedas

ηr =

(2√

2 ERk2r20

eV

)1/4

, (32)

wherek is the wavevector andER the recoil energy as definedin (19), and we have neglected thecos(θ) term for simplicity.

5In practice it is advantageous to avoid exact cylindrical symmetry in orderto have all three vibrational frequencies distinct, but this will unnecessarilycomplicate the present discussion.

The significance of (32) is that, for a given ion and wavevec-tor, the Lamb–Dicke parameter of the radial confinement isdictated primarily by the choice of electrode size (r0) and rfvoltage amplitudeV. The required rf frequencyΩV is dictatedbyV/r2

0 through (30) and the stability conditionq' 1/√

2.Recall from the discussion of the switching rate, Sect. 3.4,

that we want the Lamb–Dicke parameter for theaxial mo-tion to be around 1, assuming there is only a small numberof ions in the trap. We also want the ions to adopt theshape of a linear string, so the radial confinement must betighter than the axial confinement [see (11), (12)]. Takentogether these two considerations imply that the Lamb–Dicke parameter for theradial motion should be much lessthan 1.

Let us now add to the linear trap an axial dc potential, sothat the ions are confined in all three dimensions, and withno axial micromotion. The most obvious way to do this is toadd positively charged electrodes to either end of the lineartrap, but this introduces a difficulty in correctly balancing therf potential so that there is no residual axial rf component. Aningenious way around this is to split the linear electrodes ofthe radial quadrupole field and impose a potential differencebetween their two ends, as described in [59]; see Fig. 1. Ineither case, the dc potential near the centre of the trap willtake the form of a harmonic saddle-point potential

φdc(x,y,z) =Uz

z20

[z2− 1

2

(x2 +y2)] , (33)

whereUz is the potential on each electrode, andz0 is a pa-rameter that is a measure of the electrode separation (its exactvalue depending on the geometry). From this equation we ob-tain the vibrational frequency for the axial harmonic motionof a trapped ion:

ωz =

√2eUz

Mz20

. (34)

This is also the frequency of the lowest mode of vibration ofa string of trapped ions (centre-of-mass mode), as discussed inSect. 3.1. The Lamb–Dicke parameter of the axial confinementis

ηz =

(ERk2z2

0

4eUz

)1/4

. (35)

Owing to Earnshaw’s theorem, it is impossible to apply anaxial dc potential without influencing the radial confinement.The dc potentialφdc(x,y,z) has the effect of expelling the ionsin the radial direction. In the presence of both static axial andfluctuating radial electric potentials, the secular (i.e. slow)radial motion is still harmonic, but the vibrational frequencyis no longerω r but [32, 59]

ω ′r =(ω2

r−12ω2

z

)1/2. (36)

However, as long asω r ωz, which is the case we areinterested in, thenω ′r 'ω r , so the previous discussion ofthe radial confinement remains approximately valid, and inparticular the stability conditionq<∼ 0.9 is not greatlychanged. The depth of the trap (and hence the ease of catch-ing ions) is given approximately by the smaller ofeUz andeV/11.


636

Element, Natural Nuclear Hyperfine λ λ Recoilisotopes abundance spin splitting S–P S–D energy

(%) (~) (GHz) (nm) (nm) (kHz)

Be 9 100 3/2 1.25001767 313 226

Mg 24 79 0 280 10625 10 5/2 1.7887631

Ca 40 97 0 397 730 3043 0.14 7/2 3.25560829

87 7 9/2 5.00236835Sr 88 83 0 422 674 12.7

135 6.6 3/2 7.18334024137 11 3/2 8.03774167

Ba 138 72 0 493 1760 5.94

199 17 1/2 40.507348201 13 3/2 30.16

Hg 202 30 0 194 282 26.6

171 14 1/2 12.6428121173 16 5/2 10.4917202

Yb 174 32 0 369 411 8.42

6 7.5 1 3.0018Li ∗ 7 92.5 3/2 11.8900 539 94.7

Table 1.List of candidate ions for information pro-cessing. Only singly-charged ions are considered,although some ions of higher charge may also beinteresting. For each element, only the most abun-dant isotope, and those having non-zero nuclearspin are shown. Unstable isotopes are not shown,although most elements in the list (all but Mg andLi) have further isotopes of half-life longer thanone week. Thalium and indium are omitted sincetheir ground states haveJ = 0 and so lack hyper-fine structure. The hyperfine splittings are for theground state in all but helium-like lithium; theyare taken from G. Werth in [30], and from [76].The S–D wavelength is only shown when the Dlevel lies below the P level. The recoil energy isbased on the S–P wavelength. For Li the S,P,Dlabels do not apply; the transitions are from themetastable triplet state. Note that the fine structuresplitting in the excited state (not shown) is alsorelevant to the Raman transition rate (see text)

6 Candidate ions

Table 1 gives a list of ions suitable for information processing.The list consists of ions whose electronic structure is suffi-ciently simple to allow laser cooling without the need for toomany different laser frequencies. The list is not intended to beexhaustive, but contains most ions that have been laser cooledin the laboratory.

For information processing, a large recoil energy is attrac-tive from the point of view of allowing a faster switching rate(22), but makes the Lamb–Dicke regime harder to achieve[see (32), (35)]. The choice of rf rather than optical transi-tions for information processing appears so advantageous asto be forced upon us. Since we require at least three long-livedlow-lying states of the ion (the states|0〉, |1〉 and|aux〉), thisimplies that the existence of hyperfine structure (i.e. a non-zero nuclear spin isotope), although it complicates the coolingprocess, may be advantageous. Indeed, for alkali-like ions(such as singly charged ions from group 2 of the periodictable), the electronic ground state isJ = 1/2, so if the nucle-ar spin is zero there are only two long-lived internal states(the Zeeman components|J,M〉= |1/2,±1/2〉), which is notsufficient. Having said this, we are not necessarily forced tochoose|aux〉 to be a third internal state of the ion (i.e. thechoice implied by Fig. 5). One could make use of the secondnormal mode of oscillation of the ion string instead, choosing|aux,0〉= |F1,M1〉⊗ |0,1,0, . . .〉 [cf equation (13)]. Recently,Monroe et al. [46] have shown that the need for an auxilliarylevel can be avoided altogether.

The other major consideration is the difficulty in gen-erating the light required for cooling and informationprocessing.

Examining Fig. 7 and Table 1, we see that9Be is an at-tractive choice, in that it allows the fastest switching rateand it requires only one laser wavelength for cooling, andthe hyperfine splitting frequency of1.25 GHz is accesible toelectro-optic modulators. However, the wavelength of313 nm

100 200 300 400 500 60010

0

101

102

Strong transition wavelength (nm)

Re

coil

en

erg

y (h

kH

z)

Be

Mg

Ca

Sr

Ba

In

Yb

LiNa

Na

HgTl

**

*

Fig. 7.The recoil energies and main (typically S–P) transition wavelengths forions that may be amenable to quantum information processing (cf Table 1).A high recoil energy is advantageous for a high switching rate, but tendsto be associated with a short wavelength. A rough rule is that the shorterthe wavelength, the more complicated and therefore less stable is the lasersystem. The starred symbols are singly ionised ions from group 1, in whicha metastable manifold is used for computing, making them unattractive in thelong term

requires the use of a dye laser (frequency doubled) which isdisadvantageous. The next most promising candidate appearsto be 43Ca. It requires two laser wavelengths for cooling,397 (or 393)nm and 866 (or 850)nm (Fig. 8), but bothcan be produced by diode lasers (one frequency doubled),which makes this ion very attractive (strontium has simi-lar advantages). Diode lasers can be made very stable inboth frequency and power. If more laser power is neededthan is possible with diode lasers, then a titanium–sapphirelaser can be used, which is also advantageous compared with


637

4 2S1/2

866.2

850.0

854.4

4

3

3

4

4

3

2

5

3.225608286 GHz

G = 23 MHz

t = 1.1 s

t = 1.05 s

4 2P1/2

4 2P3/2

3 2D5/2

3 2D3/2

393.4

396.9

Fig. 8. Low-lying energy levels of the43Ca+ ion, of electronic structure1s22s22p63s23p6nl. The hyperfine structure of theD states is omitted to keepthe diagram uncluttered. The other levels are labelled by the total angularmomentum quantum numberF (nuclear spinI = 7/2). The Zeeman sublevelsare shown for the ground state hyperfine manifold, and a possible choice ofcomputational and auxilliary levels is shown by the thickened sublevels. Anexample Raman transition is shown, for use both in sideband cooling and forthe operation of quantum gates. The3D levels are shown because they areinvolved in the state preparation (including cooling) and measurement, bothof which are an integral part of a complete quantum ‘computation’. The levelseparations and lifetimes are taken from references [62]

dye lasers. The hyperfine splitting of3.26 GHz is accessi-ble to electro-optic modulators, though less easily than thesmaller splitting in beryllium. The obvious difficulty in work-ing with 43Ca is that it is a rare isotope, having a naturalabundance of only0.14% or 1 part in 700, making an isotopi-cally enriched sample that much more expensive. However,one could carry out preliminary experiments using the97%abundant40Ca.

For a group-2 ion, the internal states required for in-formation processing, discussed in Sect. 3.2 and illustratedin Fig. 5, will be taken from the ground state hyper-fine manifold. For 43Ca, for example, one might take|F1 = 4,M1 = 4〉, |F2 = 3,M2 = 3〉 and|Faux = 4,Maux = 2〉.This choice is highlighted in Fig. 8. The degeneracy betweenthe first and auxilliary levels is lifted by an imposed magneticfield of order0.1 mT.

6.1 Example: the43Ca+ ion

To estimate laser power requirements, we will calculatethe intensity required to saturate the4S1/2–4P3/2 transi-tion in Ca+ (for laser cooling purposes) and that requiredfor Raman transitions in the ground state via a quasi-resonance with this transition (for information processingpurposes). With a two-level model for the allowed elec-tric dipole transition, the saturation intensity (defined asthe intensity giving a Rabi frequency equal to the FWHM(full width, half maximum) linewidthΓ divided by

√2 ) is

IS = 4π 2~cΓ /6λ3 = 48 mW/cm2 (using Γ = 2π ×23 MHz,λ = 397 nm). To initiate laser cooling, this intensity must beavailable in a laser beam wide enough to intersect a signifi-cant proportion of a ‘hot’ ion’s trajectory in the trap. Witha beam diameter of 1 mm, the required laser power is of order0.5 mW, which is a large overestimate in practice.

Raman transitions from|0〉 to |1〉 via a near-resonancewith an excited state|e〉 can be modelled as transitions inan effective two-level system, in which the effective Rabifrequency of the Raman transition is

Ωeff =Ω0Ω1

2∆, (37)

whereΩ0 andΩ1 are the Rabi frequencies of the single-photontransitions from levels|0〉 and|1〉 to |e〉, and∆Ω0,Ω1 is thedetuning from resonance of both of these transitions. If level|e〉 decays only to levels|0〉 and|1〉, the single photon Rabifrequencies can be obtained from the laser intensityI and thelinewidth Γ of the excited state, leading toΩeff ' IΓ 2/8IS∆.During a Raman transition, the average population of the ex-cited state|e〉 is∼Ω2

0/4∆2, and to produce, for example, a2πpulse, the pulse duration is2π/Ωeff. Therefore the probabil-ity of an unwanted spontaneous emission process such anoperation is

pem =π Γ∆

. (38)

An interesting possibility, which has not yet been tried inan ion trap, is to use the Argon ion laser line at 488 nm todrive Raman transitions. In this case, we have∆ ' 6×106Γ ,so pem' 5×10−7, allowing a million computing operationsbefore spontaneous emission is a problem. The laser intensityrequired to obtain~Ωeff = ER, if we assume equation (37)applies, is then

I = 8IS∆Γ

ER

~Γ' 2.9×109 W/m2 . (39)

To address a single ion, the laser is focussed to a tight spotof diameter∼ 10µm, so the required power is modest, oforder 0.3 W. Although the main features of this argumentare correct, in fact the situation is more complicated sincethe ion is not really the three-level system implied by (37).A complete analysis must take account of all the fine andhyperfine structure.

Let the fine structure splitting in the excited state be∆Efs,and let the hyperfine structure in the excited state be of order∆Ehfs. To calculate the Raman transition’s effective Rabi fre-quencyΩeff, we must replace (37) by a sum over the transitionamplitudes for all available transition routes. Destructive inter-ference between these amplitudes can result in a much reduced


638

effective Rabi frequency, as described in [60]. Another way ofunderstanding this is to note that if the detuning∆ ∆Ehfs/~,then the nuclear and electron angular momentaI andJ may beconsidered decoupled, to a first approximation, and the elec-tric dipole transition matrix element only couples toJ, so only∆MI = 0 transitions are allowed. A similar argument appliesto the LS coupling when∆ ∆Efs/~, in which case only∆MS= 0 transitions are allowed. Unfortunately, this meansthat forL = 0 ground states, which is the usual case, Ramantransitions from one ground state hyperfine level to anothercannot be driven if the detuning∆ is increased much beyond∆Efs/~ [61]. We conclude that a large fine structure is advanta-geous. For calcium,∆Efs/~' 2π ×7700 GHz, and choosing∆ equal to this splitting givespem' 10−5.

To confineCa to the Lamb–Dicke regime of the393 nmradiation, we requireωz = ER/~' 2π ×29 kHz. This is a rea-sonable choice for information processing if we bear in mindthe remarks made in Sect. 3.4 about off-resonant transitions.Choosing an axial electrode separation of∼ 10 mm, the volt-age required on the axial electrodes isUz∼ 0.74 volts. This isa low value compared to that typically used in ion traps. Thefact that Lamb–Dicke confinement is achieved even with sucha relatively weak trap comes from the property that the ionsare cooled to the ground state of the axial motion, which herecorresponds to a temperature small compared to the recoillimit ER/kB ' 1.4µK. The low value ofUz shows that con-tact potentials will certainly be a problem, and one must beable to compensate them by seperately controlling the voltageon each electrode. To make the radial confinement ten timesstronger than this axial confinement, we require an alternat-ing voltage on the radial quadrupole electrodes of frequencyΩV ' 2π ×1.2 MHz [see (31)] and amplitudeV ' 9 volts [see(30)], assuming a distance of∼ 1 mmfrom the axis to the ra-dial electrode surfaces. In practice there is no great difficultyin generating voltages up to several hundred volts at frequen-cies in the region of tens of MHz. Since there is no reason notto have strong radial confinement, and some advantages, suchas reduced collisional heating and loss, these larger parametervalues should be used.

7 Performance limitations

Having begun in Sect. 3.2 with an idealised treatment, in whichwe assumed operations could be carried out in an ion trap witharbitrary precision, we were more realistic in Sect. 6. It nowremains to discuss the limitations on the performance of theion trap system for information processing purposes.

Two important figures of merit for a quantum informationprocessor are the number of stored qubits, which so far in thispaper has been the number of trapped ionsN, and the num-berQ of elementary operations that can be carried out beforedissipation or decoherence causes a significant loss of quan-tum information. To a first approximation, we may quantifydissipation or decoherence by a simple rateΓ d, in which caseQ= R/Γ d, where the switching rateR is given in Sect. 3.4.If we model decoherence as if each ion were independentlycoupled to a thermal reservoir, leading to a phase decoher-ence rateγ for any individual ion, then we must takeΓ d = Nγsince the quantum computation is likely to produce entangledstates in which the off-diagonal elements of the density ma-trix decay at this enhanced rate [12, 13, 16, 17]. The origin of

the factorN here is quite simple to understand. In a state suchasa|000〉+b|111〉 the phase relationship betweena andb islost if any oneof the three qubits dephases, so the rate for de-phasing of the joint state is three times the single-qubit rateif the qubits dephase independently. It is just like in classicalprobability where if something can happen inN equally like-ly different ways, it isN times more likely to happen. Whenthrowing a dice, we are three times more likely to get an evennumber than to get a one. However, decoherence of a many-ion state in an ion trap is not yet sufficiently well understoodto tell whether such a model applies [63]. Two possible ther-mal reservoirs affecting the ion trap are electrical resistancein the electrodes and thermal radiation.

The major problems in an ion trap are spontaneous tran-sitions in the vibrational motion, i.e. heating (a random walkup and down the ladder of vibrational energy levels), ther-mal radiation (driving internal rf transitions in the ions), andexperimental instabilities such as in the laser beam power, rfvoltages and mechanical vibrations, and fluctuating externalmagnetic fields [44]. The instabilities contribute to the heatingand also imply that a laser pulse is never of exactly the rightfrequency and duration to produce the intended quantum gate.A ‘decoherence rate’ of a few kHz was quoted by Monroe etal. [44], consistent with the heating rate of 1000 vibrationalquanta per second quoted in their earlier work [37]. With theswitching rate of order20 kHz, they obtainedQ' 10 withN = 1. A heating rate of 6 quanta per second was reported byDiedrich et al. [36].

A useful model of the motion of a trapped ion is a series LCcircuit shunted by the capacitance of the trap electrodes [63,64]. The inductance in the model is given byl ' Mz2

0/Ne2

where z0 is of the order of the axial electrode separation.A resistancer is due to losses in the electrodes and otherconductors in the circuit. This resistance both damps and heatsthe ionic motion with time constantl/r, leading to a heatingrate in vibrational quanta per second [63]:

Γ heat'rl

kBT~ωz'

rNe2kBT

Mz20~ωz

. (40)

For example, substituting the parameters from Sect. 6.1, andusing r = 0.1 ohm, T = 300 K, we obtainΓ heat= 5 s−1. Itshould be borne in mind that one can only consider (40) to ap-ply once other sources of electrical noise, such as rf pickup,have been reduced sufficiently, so one cannot hope to improvethe performance merely by increasing the electrode separationz0 and voltageUz.

It is a simple matter to combine (20), (34), (37), and (40) inorder to obtainQ = R/Γ heatas a function of the experimentalparameters. However, this does not bring much insight and it isbetter to think in terms of the switching rate and decoherencerate. If we takeN = 10 ions withωz = ER/~= 2π ×29 kHz,as suggested in the previous section, the switching rate isabout1 kHz, andQ∼ 200 where we use the value just quotedfor Γ heat. These parameters indicate what will probably beachievable in the next few years.

It is not hard to show that the influence of spontaneousemission of photons by the ions in the trap is much less im-portant than the severe experimental problems just mentioned.Spontaneous emission takes place during the application ofa laser pulse, because of the unavoidable weak excitation ofan excited state of the relevant ion, as noted in the previous


639

section. (It was already remarked that spontaneous emissionbetween laser pulses is negligible, owing to the adoption ofthe ground-state hyperfine manifold for computing.) With theprobabilitypem from (38), the number of operations that canbe carried out before spontaneous emission plays a signifi-cant role isQ' 1/pem, which can be of the order of105, asremarked after (38).

The conclusion is that for the moment the limitations of theion trap are associated with the vibrational degrees of freedom,and with experimental instabilities. It is here that experimentaland theoretical work must concentrate if progress is to bemade. It remains misleading at present to talk of quantum‘computations’ taking place in the lab.

7.1 Error correction

Although it is important to build an information processor withas much precision and stability as possible, in the longer termthe aim of significant computations is almost certainly unre-alisable without something that goes beyond such ‘passive’stabilisation. The inherent instability of quantum computingwas stressed in Sect. 1. It was initially thought that anythinglike active stabilisation of a quantum computer would be im-possible, since it would rely on a means of monitoring thequantum state of the computer, which would irreversibly de-stroy the computation. However, the union of informationtheory with quantum mechanics has lead to another power-ful concept, that of quantum error correction [65–67]. Theessential idea is thatK qubits of quantum information in thequantum computer can be stored (‘encoded’) in a carefullychosen way amongN > K two-state systems. The simplestcase is to store a single qubit in three two-state systems, asfollows:

|0L〉 ≡ (|000〉+ |011〉+ |101〉+ |110〉)/2|1L〉 ≡ (|111〉+ |100〉+ |010〉+ |001〉)/2

(41)

All we have done here is written down two orthogonalstates, calling them|0L〉 and |1L〉. With two states we havea single logical qubit, K = 1, but it is stored physical-ly in N = 3 separate ions. A general state of the logicalqubit is justa|0L〉+b|1L〉. The logical qubit inhabits a two-dimensional subspace of the total Hilbert space of eightdimensions.

The computation is carried out in the specially chosen2K -dimensional subspace of the total Hilbert space (2N di-mensions) of the enlarged computer. For example, to applythe logical state rotation|1L〉〈0L |−|0L〉〈1L |, we must applyV1(−π/2) = |1〉〈0|−|0〉〈1| seperately to each ion.

The encoding is chosen so that the most likely errors causethe computer’s state to go out of the special subspace (where-as computing operations keep the state within the specialsubspace). One can detect such departures, and force the com-putation back on track,withoutcorrupting the stored quantuminformation, by making well-chosen joint measurements onthe three qubits. The exact corrective procedure is deducedfrom the error correction methods which are a central part ofclassical information theory. This is a rather subtle and beau-tiful link between classical information theory and quantummechanics. Unfortunately it would take too long to describeit fully here. A thorough discussion which does not assume

much prior knowledge is given in [66]. The above ‘simplestcase’, (41), is fully analysed there.

The encoding (41) is designed to reduce phase noise,i.e. a noise source that causes the state of each ion to pre-cess a random amount, so that|0L〉→ |000〉+eiε(θ2+θ3)|011〉+eiε(θ1+θ3)|101〉+eiε(θ1+θ2)|110〉where theθ i are random phas-es andε 1 is a small parameter indicating the level of phasenoise. A similar expression holds for|1L〉. The error termis linear in ε , since eiεθ ' 1+ iεθ . To understand the cor-rection method, re-write the quantum state in terms of thenew basis|0〉 ≡ (|0〉+ |1〉)/

√2, |1〉 ≡ (|0〉−|1〉)/

√2. Then

one finds |0L〉= |000〉+ |111〉, |1L〉= |000〉−|111〉. Now,phase noise introduces erroneous terms such as|100〉 intothe quantum state. We can detect the presence of such terms,without corrupting the logical qubit, by making measurementsthat ask not ‘what is the state of the first ion?’ but rather‘are the first two ions in the same state?’. Such a measure-ment is performed by making use of a fourth ion, whichis used as a ‘check bit’. It is prepared in the state|0〉, andthen undergoes anXOR operation with the first ion, thenthe second. Its state is finally measured in the basis|0〉,|1〉, which in practice is done by first rotating the state byV1/2(−π/2), (15), then measuring in the computational ba-sis |0〉, |1〉. The other significant measurement in this case is‘are the first and third ions in the same state?’, for whichanother check bit ion may be used, or else the first onere-used.

After the check bit measurements have been completed,one has gathered some information about the noise in thesystem, but none about the original logical state (i.e. the valuesof the coefficientsa and b). In addition, the measurementsforce the system into a state that is either noise-free, or relatedto the noise-free state by a rotationσ z≡ |0〉〈0|−|1〉〈1| of oneion. In the latter case, the measurement results indicate whichion must be rotated, so the state can be corrected by applyingV1(0)V1(π/2) =−iσ z to the relevant ion.

The above sequence of logical operations and measure-ments is not guaranteed to correct the state, but yields in thiscase a final state in which the noise terms are second orderrather than first order inε [66]. Thus, we have gained a lessnoisy final state as long asε is small. Even this simplest case isquite impressive, suppressing the noise by two orders of mag-nitude whenε ' 0.01. However, more advanced encodingsachieve an even more powerful stabilisation without great-ly increasing the complexity of the corrective procedure. Inaddition, the more general methods are not restricted to correc-tion of phase noise only. A general quantum error correctingcode (QECC) can be paramatrised in terms of the numberKof information qubits ‘encoded’, the numberN of physicaltwo-state systems used, and the degree of noise suppressionachieved. A QECC is called a “t-error-correcting” code if itcan be used to restore the encoded state after up tot of theNsystems have had arbitrary errors (i.e. state changes and entan-glement of any type and size). In practice, noise will typicallycause small errors in all the systems, rather than large errorsin a few. If the erroneous terms in the density matrix are oforderε before correction, then at-error-correcting code suc-cessfully corrects all terms up to ordert in ε , so the noise isreduced toO(ε t+1) [65–67]. The power of the method aris-es from the following, at first sight astonishing, result: the‘scale-up’N/K required to implement QEC remains boundedast→∞.


640

Quantum error correction can usefully be compared andcontrasted with the more simple ‘watchdog’ or ‘quantumZeno effect’ idea which preceded it [68]. If a system isrepeatedly measured, it will repeatedly collapse onto the meas-urement basis. This can be used to suppress any tendency ofa system to precess away from the initially measured state,and such a suppression is called the watchdog or quantumZeno effect. In the present context, the precession is causedby some ‘error’ HamiltonianHe. The Zeno effect occursif many measurements can be applied within a timet suf-ficiently small that1−|〈φ|exp(−iHet/~) |φ〉|2 ∝ t2. In sucha case, ifm measurements are made during a time inter-val ∆t, the probability the system is found in a state otherthan its initial state|φ〉 is proportional tom(∆t/m)2, whichtends to zero asm→∞. For our present purposes, we donot wish to preserve the quantum computer in a particularstate, but rather in a particular part of Hilbert space, whichcan be achieved with a generalised version of the Zeno ef-fect [69]. There is thus some similarity with quantum errorcorrection, only now if the system leaves the right part ofHilbert space, there is no corrective procedure. As a resultthe stabilisation is much less powerful than that of QEC, andis almost certainly useless for a quantum computer. This isbecause errors in a quantum computer will occur primarilyduring the action of the logic gates, and it is unlikely that themeasurement necessary for a Zeno effect in the special sub-Hilbert space could be made repeatedly during the action ofa gate.

By contrast with the Zeno effect, quantum error correc-tion allows a finite error term in the system’s density matrixto accumulate, and corrects it afterwards. This is particular-ly important to the operation of quantum gates. For example,a gate between two qubits involves a four-dimensional logi-cal Hilbert space. To allow error correction, we must ensurethat at no point is the whole action of this gate concentrat-ed into a four-dimensional physical Hilbert space. This canbe done as follows. Suppose each qubit in the QC is en-coded into two physical two-state systems. A gateU(a,b)between two such encoded qubitsa,b can then be appliedin four stepsU(a,b) = u(a2,b1) ·u(a1,b2) ·u(a2,b2) ·u(a1,b1),where the operatorsu are gates between a pair of two-statesystems, anda1,a2,b1,b2 are the sets of two-state sys-tems storing qubitsa andb. The important point is that errorcorrection can be appliedbetweenthe fouru operations. At nostage is any quantum information stored in a physical Hilbertspace only just large enough to hold it, neither is any gateU carried out in a single step. The proper combination ofthese features so as to allow stabilisation has been dubbed‘fault tolerance’ [72–74]. It is found that overall stability canbe achieved even when every operation is noisy, includingthose involved in the corrective procedure. This is under activeinvestigation.

Quantum error correction was initially discussed witha general model of error processes in the quantum computer, inorder to show that almost any imaginable error process mightin principle be corrected by such techniques [65, 66, 70, 71].Subsequently, a technique specifically adapted to the vibra-tional noise in an ion trap was proposed [75]. In this proposal,the Hilbert space is enlarged by making use of four internalstates in each ion to store each qubit in the quantum com-putation. This enables a two-qubit gate to be carried out infour steps as outlined above. Next, we require a method of de-

tecting and correcting the most likely errors in the vibrationalstate. This is done by using the first (n = 0) and fourth (n = 3)vibrational states,|0,0,0, . . .〉 and |3,0,0, . . .〉, instead of thefirst two, to store the ‘bus’ qubit. The vibrational quantumnumbern is measured whenever it should be0, by swappingthe phonon state with the state of additional ions introducedfor the purpose, and probing them. Ifn is found to be1,then corrective measures are applied based on the assumptionthat a single jump upwards fromn = 0 occured, the detailsare given in [75]. Ifn is found to be2 or 4, then correctivemeasures are applied based on the assumption that a singlejump down or up fromn = 3 occured. Ifn is found to be0as it should be, a corrective measure is still required to al-low for the difference between such conditional evolution andthe unitary evolution without jumps. This procedure enablessingle jumps up or down the ladder of vibrational levels tobe corrected. Since these will be the most likely errors (ata sufficient degree of isolation from the environment), theeffect overall is to stabilise the QC. In this case the figureof merit Q is roughly squared (whereQ counts the possiblenumber of logical gatesU, not subgatesu)—a remarkableenhancement.

The above procedure makes allowance for the fact thaterrors cause the vibrational state to explore a Hilbert spaceof more than two dimensions, so in the language of quan-tum information, the bus size is larger than a single qubit,though the bus is still only used to store a single logical qubit.The bus could be made larger still by using higher excitationsof the fundamental normal mode, or by using higher-ordernormal modes. This should allow more powerful error cor-rection, and hence further increases inQ. The basic theory oferror correction gives hope that such increases inQ can bedramatic [65, 66, 72].

Error correction should not be regarded as a device merelyof interest to quantum computers. Rather, it is a powerfulmethod of enforcing coherent evolution on a quantum systemthat would otherwise be dissipative. Such a capability maybe useful for quite general situations in which stability isimportant, for example in low-noise electronic circuits, andfrequency standards. This may prove to be an area in whichquantum information theory has provided a useful tool forother branches of physics.

8 Conclusion

Let us summarise the main avenues for future work involvingthe ion trap quantum information processor.

One of the basic aims of quantum information theory is tolink abstract ideas on the nature of information with the lawsof physics. The ion trap provides a means of establishing thislink in a complete and concrete way. This will set the theoryon a more firm basis.

An important task in the theory of quantum computationis that of identifying efficient multiple-qubit quantum gates.So far, it has been assumed that the efficient gates are thosethat can be divided into a set of sufficiently few two-qubitgates. However, a system like the ion trap may allow par-ticular unitary transformations to be carried out efficientlywithout dividing them into many two-qubit operations. Aninvestigation of this should be fruitful.


641

The principle of operation of the ion trap that we have de-scribed made use of various approximations whose influenceon long quantum computations has yet to be analysed. For ex-ample, any given laser pulse on an ion will involve off-resonantstimulation of transitions other than the specific transition thepulse is designed to drive. Such effects may be unwanted, buttheir influence is unitary and accurately predictable. It wouldbe interesting to investigate whether these effects can be takeninto account in designing the sequence of laser pulses, so thatthey do not need to be corrected, or whether we are forced toregard them as errors.

Quantum computation will certainly require error correc-tion if it is ever to be useful for computational purposes. Theion trap provides a guide to the specific type of error correc-tion likely to be required in the future. The basic tools of errorcorrection are now fairly well understood, but there is muchwork to be done in bringing them to bear on the ion trap. In ad-ditition, these ideas may offer significant advantages for otheruses of the ion trap, such as frequency and mass standards;this should be explored.

Error correction only works once the level of noise inthe trap is brought sufficiently low by careful constructionand isolation. Experimental ion trap systems must be mademuch more stable than they are at present before they can takeadvantage of error correction of multiple errors among manyqubits. This is not just a question of technology, but also ofa better understanding of the noise processes, especially theinfluence of electrical noise in the electrodes providing axialconfinement. The most immediate experimental challenge isto cool a many-ion crystal to the motional ground state.

Acknowledgements.I would like to acknowledge helpful conversations withR. Thompson, D. Segal, D. Stacey and especially J. Brochard. C. Monroe andD. Wineland provided very useful comments on the manuscript, and D. Jameshelped in finding atomic data. I am supported by the Royal Society.

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